Reviews in Computational Chemistry Volume 18

152 Charge-Transfer Reactions in Condensed Phases
where nabs and nem are the first spectral moments for absorption and emission,
respectively:
Ð
nIabs=emðnÞdn
nabs=em ¼ Ð
Iabs=emðnÞdn
½7Š
Here Iabs=emðnÞ is the transition intensity. The vibrational reorganization
energy lv is defined in terms of force constants, ka, and displacements,
Qa, of the vibrational normal coordinates Qa as lv ¼ 1 P
2 a ka Q2 a .15–17 In
this chapter, we use l for the solvent component of the classical reorganization
energy lcl. The subscripts 1 and 2 are used to distinguish between the reorganization
energy of the initial (i ¼ 1) and final (i ¼ 2) ET states when the reorganization
energies in these states are different.
The mean of the first two moments gives the equilibrium free energy
difference between the final and initial states of the ET reaction
hnm ¼ 1
2 h n ð abs þ nemÞ
¼ F0 ¼ F02 F01 ½8Š
The two parameters, l cl and F0, actually fully define the parabolic ET free
energy surfaces FiðXÞ in the MH formulation (Figure 2). Calculation of these
two parameters has become the main historical focus of the ET models addressing
the thermodynamics of the ET activation barrier. The latter, according to
MH theory, can be written in terms of F0 and l cl as
F act
i ¼ ðlcl F0Þ 2
½9Š
4lcl where i ¼ 1 and ‘‘þ’’ refer to the forward transition, and i ¼ 2 and ‘‘ ’’ refer
to the backward transition.
The second spectral moments of absorption and emission lines
s 2 abs=em ¼
are equal in the MH formulation
Ð n 2 Iabs=emðnÞdn
Ð Iabs=emðnÞdn
s 2 abs ¼ s2 em
n abs=em
They are related to the classical and vibrational reorganization energies as follows
18
s 2 abs=em ¼ 2kBT l cl þ hnvlv
where kB is the Boltzmann constant and T is temperature.
2
½10Š
½11Š
½12Š